How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?

# What are we learning ?

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Competency in Focus: Calculating the limit of the function  This problem from I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016 is based on simple manipulations and limit of a function .[/et_pb_text][et_pb_text _builder_version="4.1" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#ffffff" background_color="#0c71c3" width="100%" custom_margin="48px||48px||false|false" custom_padding="20px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_color_all="#0c71c3"]

# Next understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" hover_enabled="0" border_radii="on|100px|100px|100px|100px" box_shadow_style="preset3" box_shadow_color="#ffffff"]Let $f:R \to R$ be a non-zero function such that $\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}$ exists for all y >0. Let $g(y)= \lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}$ .If g(1)=1 ,then for all y>0 ? [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.1" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.1"]
I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016. Obective Problem no. 21.
[/et_pb_accordion_item][et_pb_accordion_item title="Key Competency" _builder_version="4.1" open="off"]Limit of a function [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" open="off"]7 out of 10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.22.4" open="off"]Elementary Number Theory by David M. Burton

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="4.1" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" custom_padding="|||25px||" hover_enabled="0"][et_pb_tab title="Hint 0" _builder_version="4.1"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.1" hover_enabled="0"]

So , what we have to find in general the value of  g(y) in terms of y  given that $g(y)=\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}$ provided $g(y)=\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}$ exists .[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.1" hover_enabled="0"]

See what can we do is that g(y) can be written as follows , $g(y)=\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}= (y^3)\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3y^3}$ [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.1" hover_enabled="0"]  Again from the given condition we have $g(1)=\lim_{x \rightarrow \infty} f(x) /x^3=\lim_{xy \rightarrow \infty} \frac{f(xy)}{x^3y^3}=1$.[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.1" hover_enabled="0"]Therefore , $g(y)=(y^3)\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3y^3} =y^3$ by previous argument .  [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

# Connected Program at Cheenta

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

# Similar Problem

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